Asymptotic Analysis and Singular Perturbation Theory

where
w = − t2/3v .
31/3 It follows that as t → ∞ with t2/3v fixed,
u(x, t) ∼ 2π
(3t) 1/3
f(0)Ai − t2/3v 31/3
.
Thus the transition between oscillatory and exponential behavior is described by
an Airy function. Since v = x/t, the width of the transition layer is of the order
t 1/3 in x, and the solution in this region is of the order t −1/3 . Thus it decays more
slowly and is larger than the solution elsewhere.
Whitham [20] gives a detailed discussion of linear and nonlinear dispersive wave
propagation.
3.5 Laplace’s Method
Consider an integral
I(ε) =
∞
−∞
f(t)e ϕ(t)/ε dt,
where ϕ : R → R and f : R → C are smooth functions, and ε is a small positive
parameter. This integral differs from the stationary phase integral in (3.9) because
the argument of the exponential is real, not imaginary. Suppose that ϕ has a global
maximum at t = c, and the maximum is nondegenerate, meaning that ϕ ′′ (c) < 0.
The dominant contribution to the integral comes from the neighborhood of t = c,
since the integrand is exponentially smaller in ε away from that point. Taylor
expanding the functions in the integrand about t = c, we expect that
I(ε) ∼
Using the standard integral
we get
1
[ϕ(c)+ f(c)e 2 ϕ′′ (c)(t−c) 2 ]/ε
dt
∼ f(c)e ϕ(c)/ε
∞
−∞
∞
−∞
1 −
e 2 at2
2π
dt =
a ,
2πε
I(ε) ∼ f(c)
|ϕ ′′ 1/2 e
(c)|
ϕ(c)/ε
e 1
2 ϕ′′ (c)(t−c) 2 /ε dt.
as ε → 0 + .
This result can proved under suitable assumptions on f and ϕ, but we will not give
a detailed proof here (see [17], for example).
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